Let and be a continuous and increasing function. Prove that
and this finishes the exercise.
Let be a Riemann integrable function. If forall rationals of the interval then prove that .
Let be a square matrix and for that it holds that
Prove that is symmetric.
Taking transposed matrices back at we get that
On the other hand it holds that
Of course it holds that if a matrix is symmetric or antisymmetric and then . The proof is left as an exercise to the reader.
We can safely conclude using the above observations that for our matrix it holds that
But then for the matrix it holds that
and since the matrix is antisymmetric we conclude that and thus . Hence the result.
For any permutation define its displacement as
What is greater: the sum of displacements of even permutations or the sum of displacements of odd permutations? The answer may depend on .
Let that are diagonizable in . If and then prove that
where are the diagonial elements of and of . According to we have that
But , so and finally we conclude that: